The objective of this program is to advance understanding in algebra, group theory, representation theory and combinatorics, with particular emphasis on the mathematics underlying physics and geometry. Almost all finite dimensional semisimple Lie groups and Lie algebras occur in space-time symmetries and the development of the Standard Model of particle physics, which could not have progressed without an understanding of symmetries and group transformations. Infinite dimensional generalizations, known as Kac-Moody algebras and their associated groups, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's the class of affine Kac-Moody algebras was shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. Recently many properties of hyperbolic Kac-Moody groups and algebras have been discovered in high-energy physics, though the full mathematical structure of these objects and certain aspects of the physical settings are not well understood. Moreover, mathematical proofs of the correspondences with physical theories are lacking or incomplete, and fundamental questions about the structure of hyperbolic algebras and groups remain open. This in part serves as motivation for our current and proposed work. Fundamental questions about the structure of hyperbolic Kac-Moody groups and algebras remain unanswered. The proposal addresses these questions directly using all available mathematical techniques. Our work also involves a number of mathematical questions about hyperbolic Kac-Moody groups and algebras that are motivated by the discovery of Kac-Moody symmetry in M-theory, supergravity and dimensional reduction, solutions of multidimensional gravity and cosmological billiards. The research proposed is in line with the following long term goals: to draw parallels between Kac-Moody groups, automorphism groups of discrete and continuous buildings, Lie theory, representation theory and automorphic forms, to find a classical interpretation of hyperbolic Kac-Moody groups, to use the geometry of Lorentz space and the methods of C*-algebras and noncommutative geometry to study Kac-Moody groups, to build a mathematical framework for studying recent discoveries of Kac-Moody symmetries in physics. Our recent work and proposal incorporates a range of mathematical techniques including group theory, algebra, representation theory, analysis, geometry, arithmetic and combinatorial methods, as well as the interactions between these subjects.
The objective of this program is to advance understanding in algebra, group theory, representation theory and combinatorics, with particular emphasis on the mathematics underlying physics and geometry. We are studying infinite dimensional generalizations of the finite dimensional symmetries, known as Lie groups and Lie algebras, which occur in space-time symmetries and the development of the Standard Model of particle physics, and are widespread in their roles in diverse areas of mathematics. Infinite dimensional generalizations, known as Kac-Moody structures, naturally form two distinct classes, namely affine and hyperbolic. By the 1980's affine Kac-Moody symmetries were shown to have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. Recently many properties of hyperbolic Kac-Moody symmetries have been discovered in high-energy physics, though the full mathematical structure of these objects and certain aspects of the physical settings are not well understood. Moreover, mathematical proofs of the correspondences with physical theories are lacking or incomplete, and fundamental questions about the structure of hyperbolic Kac-Moody symmetry structures remain open. The proposal addresses these questions directly using all available mathematical techniques. Our work also involves a number of mathematical questions about hyperbolic Kac-Moody structures that are motivated by the discovery of Kac-Moody symmetry in M-theory, (the proposed unification of 5 string theories), supergravity and dimensional reduction, solutions of multidimensional gravity and cosmological billiards. Our objective is to strengthen the mathematical framework for studying these physical applications, and to identify and explore problems in algebra and geometry that are of relevance to the development of high-energy contemporary physics.
